Self-consistent approach to many-body localization and subdiffusion

An analytical theory, based on the perturbative treatment of the disorder and extended into a self-consistent set of equations for the dynamical density correlations, is developed and applied to the prototype one-dimensional model of many-body localization. Results show a qualitative agreement with the numerically obtained dynamical structure factor in the whole range of frequencies and wave vectors, as well as across the transition to nonergodic behavior. The theory reveals the singular nature of the one-dimensional problem, whereby on the ergodic side the dynamics is subdiffusive with dynamical conductivity $\sigma \left(\omega \right)\propto {\left|\omega \right|}^{\alpha }$, i.e., with vanishing dc limit ${\sigma }_0=0$ and $\alpha <1$ varying with disorder, while we get $\alpha >1$ in the localized phase.

Figure 1

High-temperature $\beta \to 0$ dynamical structure factor $S(q,\omega )$ as calculated with MCLM for $L=24,V=1$, (a) $W=0$, (b) $W=2$, and (c) $W=4$ for all $q=[0,\pi ]$.

Figure 2

(a1), (b1), (c1) Dynamical conductivity $\mathrm{Re}\sigma (q,\omega )$ and (a2), (b2), (c2) current relaxation-rate function $\gamma (q,\omega )$, obtained via MCLM on $L=24$ sites for fixed parameters $V=1$, as compared to the solution of SC equations (with effective length $L^{*}=24)$ for different $q$ and two disorders: (a) $W=1$ and (b) $W=2$.

Figure 3

(a1), (b1) $\mathrm{Re}\sigma (q,\omega )$ and (a2, b2) $\gamma (q,\omega )$ (MCLM, $L=24)$ for fixed $V=2$, few different $q$, and two disorders: (a) $W=1$ and (b) $W=2$.

Figure 4

Comparison between SC and numerical (MCLM) results of (a) $\mathrm{Re}\sigma (q,\omega )$ and (b) $\gamma (q,\omega )$ for $q=\pi /12,L=L^{*}=24$, and various disorder strengths $W=1,2,3,4$.

Figure 5

Comparison of SC solution (solid line, $L^{*}=40,\delta \omega ={10}^{-3})$ with the fit $\tilde{\sigma }\left(\omega \right)=a+b{\left|\omega \right|}^c$ (dashed line) with $c=0.8,1.0,1.5$ for $W=1.2,1.6,2.0$, respectively.

Figure 6

Optical conductivity $\sigma \left(\omega \right)$ (in a log-log scale) as evaluated from the SC theory at $V=1$ for different $W=0.8,1.4,2.0$, and various effective lengths $L^{*}=20–320$ with frequency resolution $\delta \omega ={10}^{-4}$. Insets of (b,c): scaling of $L^{*}=20$ and $L^{*}=320$ with $\delta \omega$ used in SC equations.

Figure 7

(a)–(c) Frequency resolution $\delta \omega$ dependence of $\sigma \left(\omega \right)$ for fixed effective lengths $L^{*}=640$ (left column) and $L^{*}=20$ (right column), and various disorder strengths $W=0.5,1.0,2.0$.

Figure 8

(a) Dynamical exponent $\alpha$ and (b) the inverse dielectric polarizability $1/{\chi }_d$ vs $W$ as evaluated at $V=1$ and different $L^{*}=20–320$ and $\delta \omega ={10}^{-4}$. Note that the MBL transition is determined by $\alpha =1$.